Download Foil Presentation - Poughkeepsie ACM Chapter

Secure Data Transmission: The Mathematics
Behind Identification Numbers and Check Digit
Schemes
Joseph Kirtland
Marist College
Meeting of the Poughkeepsie Chapter of the ACM
Marist College
April 18, 2016
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
US Postal Money Order
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
Universal Product Code - UPC
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
Vehicle Identification Number &
State of Washington Driver’s License Number
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
ISBN/EAN/ISBN-13
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
Credit Card Numbers
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
German Bank Note
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
Transmission Errors
728166153146
728166753146
721866153146
721866113546
728199153146
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
Common Error Patterns
Error Type
single digit error
transposition of adjacent digits
jump transposition
twin error
phonetic error∗
jump twin error
∗ For a
Form
a→b
ab → ba
abc → cba
aa → bb
a0 ↔ 1a
aca → bcb
Relative Frequency
79.1%
10.2%
0.8%
0.5%
0.5%
0.3%
= 2, 3, 4, 5, 6, 7, 8, 9.
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
Modulo Arithmetic
Let x and n be integers with n > 0. The remainder r obtained
when x is divided by n is denoted by x (mod n) where
0 ≤ r ≤ n − 1.
Let n be a positive integer. Given two integers x and y , then x is
congruent to y modulo n, denoted x ≡ y (mod n), if x and y
have the same remainder when divided by n.
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
US Postal Money Order
General Form: a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11
Document Number: a1 a2 a3 a4 a5 a6 a7 a8 a9 a10
Check Digit: a11
a11 = (a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10 )
(mod 9)
Valid Number: 67021200988
(6 + 7 + 0 + 2 + 1 + 2 + 0 + 0 + 9 + 8) (mod 9) = 35 (mod 9)
=8
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
Detection Rate
dr =
number of ways an error is detected
number of ways to make an error
Single Digit Errors (a → b): 10 choices for a and 9 choices for b
resulting in 90 possible ways.
Transposition of Adjacent Digit Errors (ab → ba): 10 choices for a
and 9 choices for b resulting in 90 possible ways.
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
Detection Rate - USPMO
Single Digit Errors:
dr =
88
= 98%
90
Transposition of Adjacent Digit Errors:
dr =
0
= 0%
90
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
UPC and EAN
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
UPC Version A
General Form: a1 − a2 a3 a4 a5 a6 − a7 a8 a9 a10 a11 − a12
a1 - number system character
a2 a3 a4 a5 a6 - company number
a7 a8 a9 a10 a11 - product number
a12 - check digit
3a1 + a2 + 3a3 + a4 + 3a5 + a6 + 3a7 + a8 + 3a9 + a10 + 3a11 + a12 ≡ 0
(mod 10)
No. System Char: 0 - General Groceries; 2 - Meat, Produce,
Weight Items; 3 - Drugs and Health Products; 4 - In-Store Items;
5 - Coupons; 6,7 - Other Items;
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
UPC Version A
ID Number: 5-02003-91562
UPC: 5-02003-91562-C
3·5 + 0 + 3·2 + 0 + 3·0 + 3 + 3·9 + 1 + 3·5 + 6 + 3·2 + C = 0
(mod 10)
15 + 0 + 6 + 0 + 0 + 3 + 27 + 1 + 15 + 6 + 6 + C = 0
(mod 10)
79 + C = 0
(mod 10)
Thus C = 1 and the UPC is 5-02003-91562-1.
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
UPC Version A - Single Digit Errors
The UPC Check Digit Scheme catches all single digit errors.
...a... → ...b...
c + 3a = 0
(mod 10)
&
c + 3b = 0
(mod 10)
(c + 3a) − (c + 3b) = 0 (mod 10)
3a − 3b = 0 (mod 10)
3(a − b) = 0 (mod 10)
a − b = 0 (mod 10)
a=b
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
UPC Version A - Transposition of Adjacent Digit Errors
The UPC Check Digit Scheme does not catch all transposition of
adjacent digit errors.
. . . ab . . . → . . . ba . . .
c + 3a + b = 0
(mod 10)
&
c + 3b + a = 0
(mod 10)
(c + 3a + b) − (c + 3b + a) = 0 (mod 10)
2a − 2b = 0 (mod 10)
2(a − b) = 0 (mod 10)
Undetected when |a − b| = 5.
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
UPC Version A - Detection Rates
Single Digit Errors:
dr =
90
= 100%
90
Transposition of Adjacent Digit Errors:
dr =
80
= 89%
90
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
Weighted Sums - Mod 10
n
X
w i ai = 0
(mod 10)
i=1
w1 a1 + w2 a2 + · · · + wn an = 0
(mod 10)
wi - weights
ai - digits in the identification number
catch all
single digits errors
catch all transposition
of adjacent digit errors
wi relatively prime
to 10
wi+1 − wi relatively
prime to 10
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
Weighted Sums in General
n
X
wi ai = 0
(mod M)
i=1
w1 a1 + w2 a2 + · · · + wn an = 0
(mod M)
wi - weights
ai - digits in the identification number
catch all
single digits errors
catch all transposition
of adjacent digit errors
wi relatively prime
to M
wi+1 − wi relatively
prime to M
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
IBM Scheme
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
IBM Scheme
(
2a
σ(a) =
2a − 9
if 0 ≤ a ≤ 4
if 5 ≤ a ≤ 9
= 2a +
2a 10
(mod 10)
0 1 2 3 4 5 6 7 8 9
σ=
0 2 4 6 8 1 3 5 7 9
General Form : a1 a2 · · · an−1 an
n-even:
σ(a1 ) + a2 + σ(a3 ) + a4 + · · · + σ(an−1 ) + an = 0 (mod 10)
n-odd:
a1 + σ(a2 ) + a3 + σ(a4 ) + · · · + σ(an−1 ) + an = 0 (mod 10)
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
IBM Scheme
0 1 2 3 4 5 6 7 8 9
σ=
0 2 4 6 8 1 3 5 7 9
Specific Number: 1324136 9
σ(1) + 3 + σ(2)+4 + σ(1) + 3 + σ(6) + 9 (mod 10)
= 2 + 3 + 4 + 4 + 2 + 3 + 3 + 9 (mod 10)
= 30 (mod 10)
=0
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
IBM Scheme - Single Digit Errors
The IBM Scheme catches all single digit errors.
...a... → ...b...
c + σ(a) = 0
(mod 10)
&
c + σ(b) = 0
(mod 10)
(c + σ(a)) − (c + σ(b)) = 0 (mod 10)
σ(a) − σ(b) = 0 (mod 10)
σ(a) = σ(b) (mod 10)
σ(a) = σ(b)
a=b
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
IBM Scheme - Transposition of Adjacent Digit Errors
The IBM Scheme does not catch all transposition of adjacent digit
errors.
. . . ab . . . → . . . ba . . .
c +σ(a)+b = 0
(mod 10)
&
c +σ(b)+a = 0
(mod 10)
(c + σ(a) + b) − (c + σ(b) + a) = 0 (mod 10)
σ(a) − a − σ(b) + b = 0 (mod 10)
σ(a) − a = σ(b) − b (mod 10)
Undetected when a = 0, b = 9 or a = 9, b = 0.
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
IBM Scheme - Transposition of Adjacent Digit Errors
0≤a≤4
5≤b≤9
σ(a) − a (mod 10)
σ(b) − b (mod 10)
2a − a (mod 10)
2b − 9 − b (mod 10)
a (mod 10)
b − 9 (mod 10)
b + 1 (mod 10)
0
9
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
IBM Scheme - Transposition of Adjacent Digit Errors
σ(0) − 0 = 0 − 0 = 0
σ(1) − 1 = 2 − 1 = 1
σ(2) − 2 = 4 − 2 = 2
σ(3) − 3 = 6 − 3 = 3
σ(4) − 4 = 8 − 4 = 4
σ(5) − 5 = 1 − 5 = 6
σ(6) − 6 = 3 − 6 = 7
σ(7) − 7 = 5 − 7 = 8
σ(8) − 8 = 7 − 8 = 9
σ(9) − 9 = 9 − 9 = 0
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
IBM Detection Rates
Single Digit Errors:
dr =
90
= 100%
90
Transposition of Adjacent Digit Errors:
dr =
88
= 98%
90
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
A Theorem by Gumm, 1985
Suppose an error detecting scheme with an even modulus detects
all single digit errors. Then for every i and j there is a transposition
error involving positions i and j that cannot be detected.
a1 · · · ai · · · aj · · · an → a1 · · · aj · · · ai · · · an
Proof:
Let 2m be the even modulus.
For a1 a2 · · · an , the CDS is
σ1 (a1 ) + σ2 (a2 ) + · · · σn (an ) = 0
(mod 2m)
To catch all single digit errors, each σi must be a permutation
of Z2m = {0, 1, 2, . . . , 2m − 1}.
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
A Theorem by Gumm, 1985
To detect all transposition errors involving positions i and j
(. . . a . . . b . . . → . . . b . . . a . . ., a 6= b), we must have
σi (a) + σj (b) 6= σi (b) + σj (a)
or
σj (b) − σi (b) 6= σj (a) − σi (a)
In this case, the map σ(x) = σj (x) − σi (x) is a permutation
of Z2m .
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
A Theorem by Gumm, 1985
Summing the elements of Z2m (mod 2m) obtains
m = m + 0 + (1 + 2m − 1) + (2 + 2m − 2) + + (m − 1 + m + 1)
X
=
x
X
=
σ(x)
X
=
(σj (x) − σi (x))
X
X
=
σj (x) −
σi (x)
=m−m
=0
This is a contradiction.
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
ISBN...ISBN-10...ISBN-13...EAN-13...EAN
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
ISBN-10
General Form: a1 a2 a3 a4 a5 a6 a7 a8 a9 a10
a1. . .: group/country number (0,1=English, 3=German,
9978=Ecuador)
ai . . . aj : publisher number
aj+1 . . . a9 : serial number
a10 : check digit
10·a1 +9·a2 +8·a3 +7·a4 +6·a5 +5·a6 +4·a7 +3·a8 +2·a9 +1·a10 ≡ 0
(mod 11)
If a10 = 10, the letter X is used.
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
ISBN-10
ISBN-10: 1-86197-876-6
10·1 + 9·8 + 8·6 + 7·1 + 6·9 + 5·7 + 4·8 + 3·7 + 2·6 + 1·6 ≡ 0
10 + 72 + 48 + 7 + 54 + 35 + 32 + 21 + 12 + 6 ≡ 0
297 = 17·11 ≡ 0
(mod 11)
(mod 11)
(mod 11)
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
ISBN-10 Detection Rates
Single Digit Errors:
dr =
90
= 100%
90
Transposition of Adjacent Digit Errors:
dr =
90
= 100%
90
But why don’t we like this scheme?
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
German Banks
German banks use a mod 11 weighted scheme that does catch all
of the errors from the first table and any transposition error when
n ≤ 10.
2a1 + 22 a2 + 23 a3 + · · · + 2n an = 0
(mod 11)
Transposition involving position i and j where 1 ≤ i < j ≤ n.
...a...b... → ...b...a...
c +2i a+2j b = 0
(mod 11)
&
c +2i b+2j a = 0
2i (2j−i − 1)b = 2i (2j−i − 1)a
(mod 11)
(mod 11)
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
German Banks
21 − 1 = 1
22 − 1 = 3
23 − 1 = 7
24 − 1 = 15 = 3·5
25 − 1 = 31
26 − 1 = 63 = 32 ·7
27 − 1 = 127
28 − 1 = 255 = 3·5·17
29 − 1 = 511 = 7·73
210 − 1 = 1023 = 3·11·31
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
Symmetries of a Regular Pentagon
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
Symmetries of the Regular Pentagon
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
Symmetries of a Regular Pentagon
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
Symmetries of a Regular Pentagon
A
A
A
2=
C
A
4=
E
A
6=
E
A
8=
C
0=
B C
B C
D E
D E
B C
D E
D E
A B
B C
A B
D E
C D
B C
D C
D E
B A
B C D E
B A E D
A
B
A
3=
D
A
5=
A
A
7=
D
A
9=
B
1=
C D E
D E A
B C D E
E A B C
B C D E
E D C B
B
C
B
C
C
B
D E
A E
B C
A E
D E
D C
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
The Multiplication Table for D5
8∗3=5
3∗8=6
∗
0
1
2
3
4
5
6
7
8
9
0
0
1
2
3
4
5
6
7
8
9
1
1
2
3
4
0
9
5
6
7
8
The operation is not commutative!
2
2
3
4
0
1
8
9
5
6
7
3
3
4
0
1
2
7
8
9
5
6
4
4
0
1
2
3
6
7
8
9
5
5
5
6
7
8
9
0
1
2
3
4
6
6
7
8
9
5
4
0
1
2
3
7
7
8
9
5
6
3
4
0
1
2
8
8
9
5
6
7
2
3
4
0
1
9
9
5
6
7
8
1
2
3
4
0
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
The Verhoeff Scheme
Number: a1 a2 · · · an
0 1 2 3 4 5 6 7 8 9
not
Permutation: σ =
0 4 3 2 1 6 7 8 9 5
unique
Any permutation works as long as
σ(a) ∗ b 6= σ(b) ∗ a
for a 6= b.
∗ is the group operation from D5 .
σ n−1 (a1 ) ∗ σ n−2 (a2 ) ∗ σ n−3 (a3 ) ∗ · · · ∗ σ(an−1 ) ∗ an = 0
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
The Verhoeff Scheme
Note: Since σ is a permutation, for each i ∈ N, σ i is also a
permutation.
0 1 2 3 4 5 6 7 8 9
σ3 =
◦
0 4 3 2 1 6 7 8 9 5
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
◦
0 4 3 2 1 6 7 8 9 5
0 4 3 2 1 6 7 8 9 5
σ3
0 1 2 3 4 5 6 7 8 9
=
0 4 3 2 1 8 9 5 6 7
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
The Verhoeff Scheme - Single Digit Errors
Catches all single digit errors.
...a... → ...b...
c1 ∗ σ i (a) ∗ c2 = 0
&
c1 ∗ σ i (b) ∗ c2 = 0
c1 ∗ σ i (a) ∗ c2 = c1 ∗ σ i (b) ∗ c2
σ i (a) ∗ c2 = σ i (b) ∗ c2
σ i (a) = σ i (b)
a=b
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
The Verhoeff Scheme - Transposition of Adj. Digit Errors
Catches all transposition of adjacent digit errors.
. . . ab . . . → . . . ba . . .
c1 ∗ σ i+1 (a) ∗ σ i (b) ∗ c2 = 0
&
c1 ∗ σ i+1 (b) ∗ σ i (a) ∗ c2 = 0
c1 ∗ σ i+1 (a) ∗ σ i (b) ∗ c2 = c1 ∗ σ i+1 (b) ∗ σ i (a) ∗ c2
σ i+1 (a) ∗ σ i (b) ∗ c2 = σ i+1 (b) ∗ σ i (a) ∗ c2
σ i+1 (a) ∗ σ i (b) = σ i+1 (b) ∗ σ i (a)
σ(σ i (a)) ∗ σ i (b) = σ(σ i (b)) ∗ σ i (a)
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
The Verhoeff Scheme
σ(σ i (a)) ∗ σ i (b) = σ(σ i (b)) ∗ σ i (a)
Since a 6= b, it follows that σ i (a) 6= σ i (b). Let c = σ i (a) and
d = σ i (b). Then
σ(σ i (a)) ∗ σ i (b) = σ(σ i (b)) ∗ σ i (a)
σ(c) ∗ d = σ(d) ∗ c
This is a contradiction.
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
The German Bundesbank Scheme
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
The German Bundesbank Scheme
Number: a1 a2 · · · a11
0 1 2 3 4 5 6 7 8 9
Permutation: σ =
1 5 7 6 2 8 3 0 9 4
A
0
D
1
G
2
K
3
L
4
N
5
S
6
U
7
Y
8
Z
9
∗ is the group operation from D5 .
σ(a1 ) ∗ σ 2 (a2 ) ∗ σ 3 (a3 ) ∗ · · · ∗ σ 10 (a10 ) ∗ a11 = 0
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
The German Bundesbank Scheme
0 1 2 3 4 5 6 7 8 9
σ=
1 5 7 6 2 8 3 0 9 4
DL0998939U1 → 14099893971
σ(1) ∗ σ 2 (4) ∗ σ 3 (0) ∗ σ 4 (9) ∗ σ 5 (9) ∗ σ 6 (8)∗
σ 7 (9) ∗ σ 8 (3) ∗ σ 9 (9) ∗ σ 10 (7) ∗ 1 = 0
5 ∗ 7 ∗ 8 ∗ 0∗1 ∗ 1 ∗ 8 ∗ 3 ∗ 4 ∗ 1 ∗ 1 = 0
0=0
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
German Bundesbank Scheme
This scheme has one major
problem, what is it?
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
Euro
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
An Error Correcting Scheme
Number: a1 a2 · · · a8 a9 a10 with a9 , a10 check digits.
a1 + a2 + . . . + a8 + a9 + a10 = 0 (mod 11)
a1 + 2a2 + . . . + 8a8 + 9a9 + 10a10 = 0 (mod 11)
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
An Error Correcting Scheme
62150334a9 a10
a1 + a2 + . . . + a8 + a9 + a10 = 0 (mod 11)
6 + 2 + 1 + 5 + 0 + 3 + 3 + 4 + a9 + a10 = 0 (mod 11)
24 + a9 + a10 = 0 (mod 11)
2 + a9 + a10 = 0 (mod 11)
a1 + 2a2 + . . . + 8a8 + 9a9 + 10a10 = 0 (mod 11)
6 + 2·2 + 3·1 + 4·5 + 5·0 + 6·3 + 7·3 + 8·4 + 9·a9 + 10·a10 = 0 (mod 11)
104 + 9·a9 + 10·a10 = 0 (mod 11)
5 + 9·a9 + 10·a10 = 0 (mod 11)
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
An Error Correcting Scheme
6215033472 → 6218033472
a1 + a2 + . . . + a8 + a9 + a10 (mod 11)
6 + 2 + 1 + 8 + 0 + 3 + 3 + 4 + 7 + 2 (mod 11)
36 (mod 11) = 3
a1 + 2a2 + . . . + 8a8 + 9a9 + 10a10 = 3i (mod 11)
6 + 2·2 + 3·1 + 4·8 + 5·0 + 6·3 + 7·3 + 8·4 + 9·7 + 10·2 = 3i (mod 11)
199 = 3i (mod 11)
1 = 3i (mod 11)
i =4
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College
References
Gallian, J. A., The Mathematics of Identification Numbers,
College Mathematics Journal 22(3), 1991, 194-202.
Gallian, J.A., Error Detection Methods, ACM Computing
Surveys 28(3), 1996, 504-517.
Consortium for Mathematics and Its Applications, For All
Practical Purposes: Mathematical Literacy in Today’s World,
9th ed., W. H. Freeman, 2013.
Kirtland, J. Identification Numbers and Check Digit Schemes,
The Mathematical Association of America - Classroom
Resource Materials Series, 2001.
Joseph Kirtland
Secure Data Transmission: The Mathematics Behind Identification Numbers and Check Digit Schemes
Marist College